Central field approximation, angular momentum and spherical functions

in the central field approximation the wave function of the stationary state of an electron in an atom will be the eigenfunction of the operators of total energy, angular and spin momenta squared and one of their projections. These operators will form the full set of commuting operators and the corresponding stationary state of an atomic electron will be characterized by total energy E, quantum numbers of orbital l and spin s momenta as well as by one of their projections. 

Formally, orbital angular momentum operator L of a particle moving with linear momentum p = —itiV at a position r with respect to some 

A vectorial product will be defined below by (5.14), and V as a tensor of first rank is defined by (2.12). Operator L may be defined also in a more general way by the commutation relations of its components. Such a definition is applicable to electron spin s, as well. Therefore, we can write the following commutation relations between components of arbitrary angular momentum j  

Because of the spherical symmetry of an atom, it is very convenient to use spherical coordinates (Fig. 5.1), defined by 

Components of the angular momentum operator are connected with the infinitesimal operators of the group of rotations in three-dimensional space [11]. 

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